Lorentz Symmetry Fractionalization and Dualities in (2+1)d

Abstract

We discuss symmetry fractionalization of the Lorentz group in (2+1)$d$ non-spin quantum field theory (QFT), and its implications for dualities. We prove that two inequivalent non-spin QFTs are dual as spin QFTs if and only if they are related by a Lorentz symmetry fractionalization with respect to an anomalous ${\mathbb{Z}}_2$ one-form symmetry. Moreover, if the framing anomalies of two non-spin QFTs differ by a multiple of 8, then they are dual as spin QFTs if and only if they are also dual as non-spin QFTs. Applications to summing over the spin structures, time-reversal symmetry, and level/rank dualities are explored. The Lorentz symmetry fractionalization naturally arises in Chern-Simons matter dualities that obey certain spin/charge relations, and is instrumental for the dualities to hold when viewed as non-spin theories.

29 pages, 3 figures. v2 appendix added and section 4.2 expanded

References in corpus (9)

• Generalized Global Symmetries
• Symmetries and Strings in Field Theory and Gravity
• A Duality Web in 2+1 Dimensions and Condensed Matter Physics
• Level/rank Duality and Chern-Simons-Matter Theories
• Classifying fractionalization: symmetry classification of gapped Z2 spin liquids in two dimensions
• Comments on One-Form Global Symmetries and Their Gauging in 3d and 4d
• Chern-Simons-matter dualities with $SO$ and $USp$ gauge groups
• Theory of Twist Liquids: Gauging an Anyonic Symmetry
• Framed Wilson Operators, Fermionic Strings, and Gravitational Anomaly in 4d

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